A Cartesian product of any finite or infinite set of copies of , equipped with the product topology derived from the discrete topology of . It is denoted . The name is due to the fact that for , this set is closely related to the Cantor set (which is formed by all numbers of the interval which admit an expansion in base 3 formed by 0s and 2s only), and this gives rise to a one-to-one correspondence between and the Cantor set, which is actually a homeomorphism. In the symbol denoting the Cantor discontinuum, can be replaced by 2 and by .

# Cantor's Discontinuum

## See also

Cantor Set
*This entry contributed by Margherita
Barile*

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## References

Cullen, H. F. "The Cantor Ternary Space." §18 in*Introduction to General Topology.*Boston, MA: Heath, pp. 77-81, 1968.Joshi, K. D.

*Introduction to General Topology.*New Delhi, India: Wiley, p. 199, 1983.Willard, S. "The Cantor Set." §30 in

*General Topology.*Reading, MA: Addison-Wesley, pp. 216-219, 1970.

## Referenced on Wolfram|Alpha

Cantor's Discontinuum## Cite this as:

Barile, Margherita. "Cantor's Discontinuum." From *MathWorld*--A Wolfram Web Resource, created by Eric
W. Weisstein. https://mathworld.wolfram.com/CantorsDiscontinuum.html